#Java, 85 bytes
This is surprisingly short in Java.
void a(int a){String b="";for(int c=0;c++<a;System.out.println(b))b+=(c>1?" ":"")+c;}
Indented:
void a(int a){
String b="";
for(int c=0;c++<a;System.out.println(b))
b+=(c>1?" ":"")+c;
}